Answer
The statement is false. To understand why, let’s delve into the properties of an isosceles triangle and the relationship between its sides.
An isosceles triangle is defined as having two sides that are equal in length, which we can denote as ‘a’, while the base (the third side) can be denoted as ‘b’. Therefore, we have two equal sides of length ‘a’ and one side of length ‘b’.
Now, the statement suggests that:
√a + √a = √b
This simplifies to:
2√a = √b
To explore this further, let’s square both sides:
(2√a)² = (√b)²
This leads to:
4a = b
While this could hold true for some specific values of ‘a’ and ‘b’, it is not a generalized statement for all isosceles triangles. In fact, for any given isosceles triangle, ‘b’ can be any value that satisfies the triangle inequality theorem.
For an example, let’s consider an isosceles triangle where both sides are 5 units (a = 5) and the base is 6 units (b = 6):
- Calculating the left side: 2√5 ≈ 4.472
- Calculating the right side: √6 ≈ 2.449
- Clearly, 4.472 is not equal to 2.449.
This illustrates that the sums of the square roots of the sides do not equal the square root of the remaining side in general cases of isosceles triangles.
In conclusion, while certain pairs of specific lengths might satisfy this relation, it is not true for all isosceles triangles. Therefore, the original statement is indeed false.